Beyond imaging: on the quantitative analysis of tomographic volume data

Claudia Redenbacha, Alexander Rackb, Katja Schladitzc, Oliver Wirjadic, Michael Godehardtc

Beyond imaging: on the quantitative analysisof tomographic volume data

Tomographic techniques are a valuable analytical tool asthey deliver 3D spatial information on a given specimen.Both computed tomography with high spatial resolutionand quantitative volume image analysis have made enor-mous progress during the last decade. In particular for ma-terials and natural science applications the combination ofhigh-resolution three-dimensional imaging and the subse-quent image analysis exploiting the fully preserved spatialstructural information yield new and exciting insights.

In this paper, field-tested and up-to-date methods for to-mographic imaging of microstructures, for processing andfor quantitatively analysing three-dimensional images arereviewed. By selected applications from materials research,we shall underline the importance of volume image analysisas a crucial step in order to go beyond the images: it allowsdetermination of spatial cross-correlations between differ-ent constituents of a specimen, investigation of orientationsor derivation of statistically relevant information such asobject size distributions. The core part of this work consists,besides the exemple application scenarios, in the processingchain, the tools and methods used.

Keywords: Microstructure; Intrinsic volumes; Local fibredirection; Correlation analysis; Microtomography

1. Introduction

Microtomography generates images of, e. g. foams, fibrecomposites, snow, lichen or soil, that yield new insightsinto these structures. This is particularly important for weakor highly porous materials whose microstructures cannot beimaged by classical techniques such as materialography in-volving serial slicing and microscopic imaging. The use ofmicrotomography enables three-dimensional (3D) visuali-sation and virtual slicing of samples and thus boosts the un-derstanding of their morphology and properties related to it.Quantitative image analysis goes one step beyond plain im-aging by providing a quantitative description of the micro-structure. Exploiting the full spatial information containedin 3D images allows, e. g. carrying out detailed directionalanalysis, estimation of particle size distributions withoutshape assumptions, or examining the 3D connectivity of astructure, to name just a few possibilities. Moreover,macroscopic materials properties such as mechanicalstrength, permeability, or acoustic absorption can be simu-lated in the discrete structure given by 3D images or in dis-

cretised realisations of geometric models fitted to the mi-crostructure.

Many image processing and analysis methods are algo-rithmically and computationally too expensive to be simplygeneralised from 2D to 3D. This is not only due to the muchhigher number of pixels to be processed but it is particularlycaused by the increase in the number of possible local pixelconfigurations. In a black-and-white image in 2D there are,e. g. 16 configurations of 2 · 2 pixels, in 3D 2 · 2 · 2 pixelsform 256 different local configurations. As a consequence,only very efficient algorithms are useful and basic imageprocessing concepts such as neighbourhoods have to be re-thought when moving on to the third dimension.

The outline of this paper is arranged around an overviewof tomographic techniques, followed by a discussion of ba-sic image processing techniques such as filtering or seg-mentation which play an important role in the preparationof images for quantitative analysis. Use cases of quantita-tive image analysis are subsequently presented, introducingappropriate methods for different kinds of materials.

2. Tomographic imaging

Initially, the acronym tomography (from the Greek wordss 'olo| [tomos] = to cut and cq '�ueim [graphein] = to draw)was solely associated with the so-called computed axialscanning tomography (CAT) scanners for which G.N.Houndsfield and A.M. Cormack were awarded the NobelPrize for Physiology or Medicine in 1979. The basic ideaof computed tomography (CT) is that the inner mass distri-bution of a specimen can be virtually reconstructed usingprojection images taken under different angles of view.Nowadays, tomographic techniques are considered in abroader manner as any imaging method delivering cross-sectional pictures of the specimen under study. Besidesnon-destructive techniques based on, e. g., penetrating ra-diation, destructive techniques like the 3D atom probe [1]or a milling focused ion-beam [2] are also frequently con-sidered as tomographic methods.

Within the framework of this article, a brief introductionto CT using penetrating radiation will be given. More pre-cisely, ionising radiation is used, i. e. hard X-rays from la-boratory or synchrotron sources. Furthermore, as high spa-tial resolutions are employed, frequently one uses the termmicrotomography (lCT) [3]. Further details, especiallyconsidering the use of alternative types of radiation suchas neutrons or electrons are published in [4]. Likewise, spe-cific techniques such as positron emission tomography

C. Redenbach et al.: Beyond imaging: on the quantitative analysis of tomographic volume data

Int. J. Mat. Res. (formerly Z. Metallkd.) 103 (2012) 2 217

aUniversity of Kaiserslautern, Mathematics Department, Kaiserslautern, GermanybEuropean Synchrotron Radiation Facility, Grenoble, FrancecFraunhofer ITWM, Image Processing Department, Kaiserslautern, Germany

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which are mainly applied in medicine are not within thescope of this article, cf. [5].

For a standard lCT scan, the sample is placed on a rota-tion stage with the detector being at a fixed position (con-trary to medical CT, where the detector is moved and thepatient remains at a fixed position). The sample is illumi-nated while rotating and the transmitted radiation is re-corded by the detector downstream from the sample. Nowa-days, area detectors are deployed to record 2D radiographicimages of the sample: lCT is also frequently called a full-field method, as opposed to scanning techniques introducedlater in this paragraph. When neglecting scattering and dif-fraction effects, the recorded images are spatial maps ofthe attenuation coefficient of the sample. More precisely,each image element (pixel) is associated with a line integralof the X-ray attenuation along the corresponding X-raybeam path. Different approaches exist to virtually recon-struct the inner mass distribution of the specimen from theprojection images taken at different angles of view [6]: theproblem can be formulated as a set of independent equa-tions, to be solved (in an iterative manner) by algebraicmethods. Alternatively, a summation of the measured inte-gral attenuation profiles in real or Fourier space can be per-formed (filtered backprojection algorithm or the directFourier inversion). The reconstruction is performed cross-sectionally slice by slice generating a complete tomo-graphic volume image consisting of a stack of such slices.

The early developments of hard X-ray lCT until the endof the last millennium were dominated by the quest for theultimate resolution. The technical developments at labora-tory or synchrotron sources established X-ray lCT as a un-ique analytical tool providing high spatial resolutions, cf.,e. g. [7]. More recent developments are focusing on novelcontrast schemes with either increased sensitivity or com-plementary contrast modes on one side, and on high dataacquisition speed towards real-time 3D tomographic imag-ing on the other.

The advantage of using laboratory X-ray sources for lCTis clearly the easy access together with the drastically re-duced costs compared to large-scale facilities. Synchrotronlight sources remain the first choice for flagship applica-tions involving either novel techniques or delicate samples,or both. Using the intense flux of synchrotron light sourcesallows increasing the contrast, more exactly the sensitivity,by introducing sophisticated methods such as holotomogra-phy [8], analyser-based imaging (cf., e. g. [9]) or grating-

based interferometry (see, e. g. [10]). Briefly, these meth-ods, frequently called phase sensitive imaging, measure di-rectly or indirectly the local electron density rather thanthe local mass density as it is the case for absorption basedX-ray imaging. The latter, which is also very dependent onthe available photon flux density, benefits from the use ofsynchrotron light sources as well: the number of photonsemitted allows pushing even further the density related ma-terial contrast [11]. In the case where polychromatic radia-tion is employed rather than (quasi-)monochromatic, afurther gain of photon flux density is reached. Conse-quently, data acquisition speed can be increased dramati-cally, allowing up to several 10000 X-ray images per sec-ond. It is thus possible to follow fast processes in 2D witha time-resolution down to the microsecond range or,equivalently, to perform full tomographic scans in the rangeof seconds [12, 13]. In contrast, monochromatic radiationand the common high photon flux density is required whenlCT is combined with classical diffraction imaging oncrystals (X-ray topography) to track grains in 3D (see, e. g.[14]). Regardless of the flux, different tomographic geome-tries are available to extract 3D images of flat objects,which generally can be considered as a challenge for classi-cal lCT due to their shape (cf., e. g. [15] and [16]).

In the last few years, scanning tomographic techniqueshave been deployed more and more frequently, as they pro-vide contrast modes complementary to the ones listedabove. Scanning a sample through the focal spot of an X-ray optical system allows, for example, collecting the fluor-escence signal and/or the powder-diffraction signal whichgives, via confocal approaches or by tomographic techni-ques, access to the local chemical species distribution, localstrain or the distribution of crystalline phases within theprobed specimen (see, e. g. [17–20]). The imaging sensitiv-ity is even further increased when ptychographic tomogra-phy is used, which also increases the demands on the coher-ence of the source [21].

Within this article, the aim is to show how to go beyondimaging, i. e. how to use the highly resolved, rich-con-trasted images as the basis for quantitative image analysis.This step is of crucial importance as most applications,e. g. in the area of materials research, not only requireimages but a sound quantitative analysis as well. As an ex-ample, a microtomographic dataset of two joined glass fibrereinforced polymer plates is shown in Fig. 1: the glass fibresare clearly visible, thanks to the high contrast (data were


218 Int. J. Mat. Res. (formerly Z. Metallkd.) 103 (2012) 2

(a) (b)

Fig. 1. Subvolumes of a glass fibre reinforcedpolymer with a welded join and estimatedlocal fibre directions. Sample: Institut fürVerbundwerkstoffe Kaiserslautern. Imaging:A. Rack, BESSY, [22]. Pixel size 3.5 lm, totalsample size approximately 7 · 7 · 3 mm3,analysed sub-volumes (1.4 mm)3, [23]. The �component of the local fibre directions in polarcoordinates is colour coded. blue correspondsto the z-axis ð� ¼ 0Þ and red to � ¼ p=2. (a) Fi-bre system far from the welding line, (b) Fibresystem close to the welding line.

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taken at the BAMline of the BESSY-II light source, Ger-many, see [22]). A common question which cannot be an-swered by plain volume rendering is, whether the orienta-tion of the fibres changes with respect to their distance tothe place where the plates are welded [23].

3. Image processing

Image processing covers the gap between image acquisitionand image analysis, i. e. its purpose is to prepare the raw im-age data for subsequent numerical measurements. Out ofthe wide range of methods for image denoising and smooth-ing, contrast enhancement, grey value equalisation, or edgedetection, only a very narrow choice motivated by the usecases presented in Section 4 is introduced here.

In general one has to bear in mind that in the 3D worldsome sophisticated methods developed for 2D images arenot applicable due to the excessive computation time ormemory consumption being required. Moreover, aiming ata quantitative analysis, everything that would bias the ana-lysis has to be avoided.

In most cases, the final processing step is segmentation,which – in general – describes the process of separating animage into disjoint areas which contain the objects orphases of interest – depending on the context. For most im-age analysis tasks, image segmentation is an essential pre-requisite, as it makes it possible to measure size, shape, ori-entation and other properties from an image.

3.1. Images and adjacency systems

In order to describe suitably our methodology, we will re-quire mathematical definitions of concepts such as images,pixels, and adjacency of pixels. Mostly, these formalisewhat one might expect intuitively. Nevertheless, precise de-finitions help to describe the methods concisely. Moreover,in 3D there are some notably unobvious concepts, the moststriking one being the choice of discrete adjacency, whichwe therefore describe below in more detail.

Let L ¼ ðs1; s2; s3ÞZ3 be a three-dimensional cuboidallattice with lattice spacings s1; s2; s3 > 0. For all image dataused in this paper we have s1 ¼ s2 ¼ s3 ¼ s, however, theanalysis methods work as well for the general setting. De-note byW � R

3 a cuboidal observation window. An imagecan be seen as a function

f : L\W ! V

where V is either the set of real or complex numbers, R orC, respectively, or V ¼ f0; . . . ; 2n � 1g with n ¼ 1; 8; 16or 32. If V ¼ f0; 1g, then f is called binary image. In thiscase the image is often described as a set X � R

3 observedat the lattice points within the observation window, that isthe intersection X \ L\W . The function f is thereby justthe indicator function IX of X, restricted to the observablepoints f ¼ IXjL\W , hence f ðxÞ ¼ 1 if x 2 X \ L\W andf ðxÞ ¼ 0 otherwise. The term pixel or voxel is used for bothx 2 L\W and the pair ðx; f ðxÞÞ. With a slight abuse of no-tation we speak of the lattice spacing s as pixel size, too.

For processing as well as for analysis, the discrete con-nectivity within the image is crucial. In 2D, the neighbour-hood of a pixel x is completely described by the edges con-necting x to its direct neighbours. In 3D, an unambiguous

description needs to be more detailed. We therefore use ad-jacency systems consisting of vertices, edges, faces, andcells, instead of neighbourhoods following [24]. LetC ¼ ½0; s�3 be the unit cell of the lattice L. Its vertices canbe written as sðv1; v2; v3Þ with vi 2 f0; 1g; i ¼ 1; 2; 3,and binarily coded by 2v1þ2v2þ4v3 .

The 256 subsets n‘, ‘ ¼ 0; . . . ; 255, of vertices of C canuniquely be indexed by the sum of the codes of their ele-ments. A local adjacency system is formed by the convexhulls F‘ ¼ conv n‘ of subsets of vertices of C fulfilling anumber of consistency conditions [25, Definition 3.2]. Atranslation of the local adjacency system into all latticepoints then yields the adjacency system. The maximal adja-cency system F26 ¼

S

x2LfF0 þ x; . . . ;F255 þ xg is formedby all subsets of vertices of C and corresponds to the 26-neighbourhood. The minimal adjacency systemF6 ¼

S

x2L F 0ðC þ xÞ [ F 1ðC þ xÞ [ F 2ðC þ xÞ [ fC þ xggenerated by the vertices F 0ðCÞ, the edges F 1ðCÞ, the facesF 2ðCÞ of C, and C corresponds to the customarily used 6-neighbourhood. The pairs ðF26; F6Þ and ðF6;F26Þ are consis-tent in the following sense: if foreground and background ofa binary image are equipped with these adjacency systems,then discrete closed surfaces divide the background into twoconnected components (Jordan surface theorem) and the Eu-ler numbers of foreground and background can be estimatedconsistently. However, if ðF26; F6Þ is used, foreground andbackground are treated to a large extend non-uniformly. Thisis particularly unnatural when foreground and backgroundare just two similar constituents of a microstructure. To over-come this drawback, Ohser et al. [26] introduced the adja-cency systems F14:1 generated by the six congruent tetrahe-dra F139, F141, F163, F177, F197, and F209, and F14:2 generatedby the six tetrahedra F43, F141, F149, F169, F177, and F212.These two adjacency systems are self-consistent, hence fore-ground and background can be equipped with the same adja-cency system without creating paradoxes. The only down-side of their use is the loss of symmetry.

The discretisation of a compact set X � R3 can be

roughly described as a replica of X, built exclusively fromthe blocks provided by the chosen adjacency system F.More precisely, the discretisation X \ F of X with respectto the adjacency system F is the union of all elements F ofF for which all vertices F 0ðFÞ hit X:

X u F ¼[

fF 2 F : F 0ðFÞ � Xg

The microstructures of materials are usually heterogeneous,thus a suitable mathematical model for them is randomclosed sets – random variables whose realisations areclosed subsets of R3. Let F be the system of closed subsetsof R3 furnished with the so-called hit-or-miss �-algebra Fgenerated by the sets fF 2 F : F \A 6¼ ;g for all compactsets A � R

3. A random closed set is then a measurable map-ping from an abstract probability space ðX; A; PÞ into thespace of closed sets:

N : ðX; A; PÞ 7! ðF ; FÞHere,X is a set, andA andP denote a �-algebra and a prob-ability measure on X, respectively. The random closed setN is macroscopically homogeneous (or stationary) if its dis-tribution is invariant with respect to translations. If the dis-tribution of N is rotation invariant, then N is called isotrop-ic. Thus macroscopically homogeneous roughly means



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that the random structure observed in a sample does onaverage not depend on the position of the sample withinthe specimen. Isotropic structures are those which are –again on average – invariant with respect to rotations ofthe sample.

3.2. Segmentation

Image segmentation is one of the most important and well-studied areas in image processing, but at the same time itcannot be considered as solved and requires specialised so-lutions in most cases. From a mathematical perspective, im-age segmentation is an inverse problem, which is frequentlyill-posed. There are many reasons for this ill-posedness, butcommon causes are insufficient resolution and noise. Bothcauses lead to uncertainties in defining the structures whichare to be characterised. Therefore, prior or expert knowl-edge needs to be incorporated into the design of specialisedsegmentation routines for most images. Due to this fact, itis beyond the scope of the present paper to give an overviewof the segmentation algorithms which have been applied totomographic images in the past. General overviews of im-age segmentation algorithms can be found, e. g., in [27–29].

One basic sub-problem of image segmentation is that ofimage binarisation. Loosely speaking, binarisation trans-forms an arbitrary image into a binary image of the samesize. The simplest and most common image binarisationprocedure is thresholding. It decides whether or not a pixelx belongs to the foreground independently for each pixelon the basis of its value f ðxÞ. Despite its simplicity, this ap-proach remains the most generally applicable binarisationmethod, and can lead to good results when combined withadequate noise reduction algorithms. What needs to be an-swered is the question of how to choose the threshold value.Here, one may differentiate between global and localthresholding algorithms, depending on whether the thresh-old value is computed for an entire image or for each pixelindividually.

One example of a widely used rule for computing athreshold value is due to Otsu [30]. Under the assumptionthat the image pixel values’ histogram has a bi-modal shape,this algorithm searches for the threshold value which bestseparates these two grey value regions in terms of inner-class grey value variance and intra-class squared distance.

Global thresholding is inapplicable if the grey valueswithin the image fluctuate on a large scale, for instancedue to beam-hardening. A remedy in these cases is a globalgrey value correction, known as shading correction from2D image analysis. The image is filtered by a mean filterusing a large filter mask and subsequently subtracted pix-el-by-pixel from the original. Even better results are oftenobtained when the mean filter is replaced by a morphologi-cal opening: a minimum filter (erosion) is followed by amaximum filter (dilation) with a filter mask of generalshape (the so-called structuring element, see [31] for de-tails).

Another segmentation task is the labelling of connectedcomponents such as grains, pores, or inclusions. Thoughbeing in principle identical to the same task in two dimen-sions, labelling becomes much more demanding in 3D.First of all, the discrete connectivity within the lattice hasto be defined unambiguously (see Section 3.1). Secondly,keeping track of equivalent labels during the actual label-

ling is much more complicated than in 2D (see [32] formore details).

For the separation of particles in mutual contact such asthe grains of a powder or for the reconstruction of cells ofan open cell foam, labelling will fail as the objects are con-nected in the binary image. A combination of the Euclideandistance transform with the watershed transform will how-ever work as long as the objects are \sufficiently spherical"and their sizes do not differ too much: Let X � R

3 be the setunder consideration. The Euclidean distance transformEDT maps each point in R

3 to its shortest distance to thecomplementary set R3nX,

EDT : R3 7! ½0;1Þ : x 7!minfjjx� yjj : y 2 R

3nXg

Clearly, EDTðyÞ ¼ 0 for all y 2 R3nX while ball-shaped re-

gions in X will have local maxima in their centres. On bin-ary images, the EDT can be calculated in linear time [33].Inversion of the EDT image EDTðX \ L\WÞ turns localmaxima into minima: f ðxÞ ¼ maxfEDTðX \ L\WÞg�EDTðxÞ. Now the watershed transform assigns a connectedregion to each local minimum. The transform can be inter-preted as the flooding of the topographic surfacefðx; f ðxÞÞ : x 2 L\Wg: water rises uniformly with grow-ing grey value f from all local minima. Watersheds areformed by all pixels where water basins filled from differ-ent sources meet. This idea was turned into an efficient al-gorithm by Vincent and Soille [34]. Finally, the image issegmented into regions and the system of watersheds divid-ing them. In order to use the sketched separation strategysuccessfully, superfluous local minima caused by discreti-sation effects, shape distortions, binarisation errors andnoise have to be suppressed. Altering the watershed trans-form so that regions smaller than a pre-defined volume areunited with neighbour regions [35] or smoothing the EDTimage using the morphological h-minima transform [31,Chapter 6] are well-tested methods for this purpose. Thelatter even allows coping with large differences in sizewhen the parameter h is adapted to the total grey value[36], [25, Section 4.2.6].

4. Quantitative analysis: Use cases

In applications, the microstructures under investigation aretypically not deterministic but show a certain degree of ran-domness. Therefore, the content of an image may be inter-preted as a random set. A wide range of geometric charac-teristics for such random sets has been established [37].There is a host of analytical methods to measure these char-acteristics from image data [25]. In practice, suitable char-acteristics as well as the methods to determine them arechosen depending on the type of microstructure: its invar-iance properties, the number of components it consists ofand whether it has a natural object structure or not.

A typical assumption is that the material under investiga-tion is a macroscopically homogeneous and isotropic ran-dom set. In this section we will discuss several analysistechniques by means of some typical use cases.

4.1. Open cell foam: Intrinsic volumes

A typical analytical task for porous media is analysing thedistributions of size and shape of the pores. This informa-



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tion can be derived from the intrinsic volumes (also calledMinkowski functionals or quermass integrals) – the vol-ume V , the surface area S, the integral of mean curvature Mand the Euler number v. For a convex object, M is, up to aconstant, its mean width. The Euler number is a topologicalcharacteristic alternately counting the connected compo-nents, the tunnels, and the holes of the particles. For a con-vex body, v = 1, for a torus v = 1 – 1 = 0, and for a spherev ¼ 1þ 1 ¼ 2. Efficient algorithms for the simultaneousmeasurement of all intrinsic volumes from 3D image dataare available from [25, Chapter 5].

Useful further characteristics such as the isoperimetricshape factors f1 ¼ 6

ffiffiffi

pp

V=ffiffiffiffiffi

S3p

, f2 ¼ 48p2V=M3,f3 ¼ 4pS=M2 can be derived from the intrinsic volumes.These shape factors are normalised such thatf1 ¼ f2 ¼ f3 ¼ 1 for a ball. Deviations from 1 thus describevarious aspects of deviation from the ball shape. Othershape characteristics are, e. g. the ratio between the volumesof an object and its convex hull characterising the convexityof the object, or the ratios between the edge lengths of theminimal-volume bounding cuboid characterising the elon-gation of the object.

An open cell foam is formed by a connected system ofstruts which are given by the edges of approximately poly-hedral cells. In order to measure geometric cell characteris-tics the pore space has to be divided into single cells usingthe cell reconstruction method described in Section 3.2.An application of this method to an open copper foam isshown in Fig. 2. The same techniques may also be used to

separate the cells in closed cell foams as well as the parti-cles in granular materials. For particles such as fibres,which have a highly non-spherical shape, the method willhowever fail.

4.2. Greenland firn: Densities of the intrinsic volumes

For components of macroscopically homogeneous micro-structures that cannot be separated into disjoint regions ina canonical way, we use the densities of the intrinsic vol-umes as basic geometric characteristics. Instead of the ab-solute values of these four functionals, their ratio to thesample volume is now considered, which yields the volumefraction VV, the specific surface area SV, and the densities ofthe integral of mean curvatureMV and the Euler number vV.For porous media, the porosity is defined as 1 – VV. A shapefactor – the structure model index (SMI) – is given byfSMI ¼ 4pVVMV=S

2V [38]. It takes values of 4, 3, and 0 for

ideal systems of non-overlapping balls, cylinders, andplanes, respectively.

As an example we analysed firn (sintered snow) samplesfrom the firn core B26 which was drilled during the NorthGreenland traverse of the Alfred-Wegener-Institut für Polar-und Meeresforschung Bremerhaven in 1995 [39]. Five firnsamples taken from different depths within the ice core wereimaged using a portable lCT scanner inside a cold room at–25 8C. From the resulting grey scale images binary imagesof the pore system of the firn were obtained by global thresh-olding. Visualisations of the firn samples are shown in Fig. 3.



(a) (b)

(c) (d)

Fig. 2. Analysis of an open cell copper foamusing cell reconstruction. Sample: Duocellfoam. Imaging: Fraunhofer ITWM. Pixel size38.15 lm, sample size approximately 24.0 ·24.6 · 24.4 mm3. (a) Slice through the imageof the strut system, (b) Slice through the imageof the reconstructed cells, (c) Visualisation ofthe foam sample, (d) Cell size distribution.

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The estimated values of the intrinsic volume densitiesand the SMI are given in Table 1. They show an increaseof fSMI with increasing depth. The starting value around 3indicates a cylindrical structure. The visualisations of thedeeper samples already show a number of isolated sphericalpores. Therefore, a further increase in fSMI towards 4 can beexpected when going deeper into the firn core.

4.3. Fibre reinforced polymers: orientation tensors

With increasing resolutions and sample sizes, the number ofsamples for which macroscopic homogeneity cannot be as-sumed rises, too. Global quantitative analysis, as shown inSections 4.1 and 4.2, provides only limited information inthese cases. Thus additional local analysis is needed. Thiscan be achieved by tessellating the sample by sub-volumes,sliding a sub-volume through the sample or assigning localmeasurement values to each pixel. Hilfer [40] defines local

porosity in each pixel as the porosity in a cubic sub-volumecentred in this pixel. This idea can of course be generalisedto all densities of the intrinsic volumes as defined in Section4.2. In this section, we present a local analysis method forfibre systems combining the locality of measurements ineach pixel with the robustness of averaging over sub-vol-umes.

Fibre systems such as the fibre component in fibre rein-forced composites, non-wovens, or fibre-reinforced con-cretes can be mathematically modelled as an anisotropic ran-domly oriented fibre system N in the 3D Euclidean space R3.The directional distribution of the fibres is crucial for materi-als properties such as the mechanical strength of fibre rein-forced composites or the filtration efficiency of a non-woven.Based on 3D image data, we want to estimate the directiondistribution RðAÞ ¼ 1=ð2pVðWÞÞE

R

W \NIAðmðxÞÞ dx, where

A is a measurable set of non-oriented directions and mðxÞ isthe direction of the fibre in the point x. The expectation E istaken with respect to the random fibre system N.



(a) (b) (c)

(d) (e)

Fig. 3. Visualisations of reconstructed tomo-graphic images of firn samples at five differentdepths. Samples and imaging: J. Freitag,Alfred-Wegener-Institut für Polar- undMeeresforschung, Bremerhaven. Pixel size40 lm, sample size approximately (16 mm3).The pore system is visualised. (a) 56 m, (b)60 m, (c) 69 m, (d) 72 m, (e) 74 m.

Table 1. Intrinsic volume densities and SMI for firn samples with different porosities.

depth(m)

porosity(%)

SV(m–1)

MV

(m–2)vV

(m–3)fSMI

56 15.71 1070.36 1.80e6 –1.21e8 2.954

60 13.16 890.00 1.57e6 –2.56e7 3.135

69 10.11 755.71 1.61e6 8.93e7 3.373

72 9.06 697.19 1.52e6 1.03e8 3.410

74 7.83 644.49 1.55e6 1.46e8 3.503

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We estimate the directional distribution R by calculatingthe local fibre direction in each pixel. Recently, several ap-proaches to local direction estimation without segmentationof the individual fibres have been developed [23, 41, 42].Here we use the method based on the Hessian matrix havingas elements the second order partial derivatives of the greyvalues, that has proved to be fast and robust for many appli-cations.

Let g� be an isotropic Gaussian smoothing kernel in R3

with parameter � > 0 adjusted to the fibre radius r > 0,� � r. Now consider the Hessian matrix HðxÞ of the secondderivatives of the image f smoothed by g�:

HijðxÞ ¼q2

qxi qxj

� �

ðf�g�ÞðxÞ; i; j ¼ 1; 2; 3; x 2 R3

The eigenvectors of H carry information about directions ofthe random field N in x. For a fibrous structure, the least greyvalue variation is expected along the fibre. Thus the eigenvec-tor corresponding to the smallest eigenvalue of the Hessianmatrix HðxÞ at x is interpreted as the local direction mðxÞ. Cu-mulation of these local direction estimates yields the volumeweighted direction distribution R as defined above.

The outer product of the components m1; m2; m3 averagedwith respect to R yields the so-called second order orienta-tion tensor haiji. That is:

aij ¼Z

mimj RðdmÞ; i; j ¼ 1; 2; 3

An estimator for the components aij is obtained by aver-aging the components of the local direction mðxÞ over smallsub-volumes W :

aij ¼X

x2W \ L\N

miðxÞmjðxÞ ð1Þ

The orientation tensors are widely used in simulation ofmaterials properties in order to incorporate anisotropic lo-cal behaviour.

A typical example of a glass fibre composite is the sam-ple shown in Fig. 4. The thickness of the fibres is constant,approximately 10 lm. The fibre weight fraction of 60%corresponds to an approximate fibre volume fraction of50%. While the contrast between fibre and matrix compo-nent is good, this high fibre volume fraction prevents seg-mentation of individual fibres. Moreover, the high densityof the sample causes a decline in the grey values towardsthe centre of the image. Consequently, global thresholdingwould erroneously imply that the local fiber volume frac-tion varies strongly. Therefore, shading correction as de-scribed in Section 3.2 is applied. Subsequently, the local fi-bre directions can be estimated and the results are in goodagreement with both visual inspection and theoretical ex-pectations: due to the injection moulding the fibres are or-iented in the filling direction near the sample boundarieswhile they lie perpendicular to this direction in the core ofthe sample.

A second example is the glass fibre composite shown inFig. 1. Very much like the previous example, the sample isheterogeneous, as sub-volumes close to the welding seambehave differently from sub-volumes further away from it.We analyse the two sub-volumes visualised in Fig. 1 usingthe Hessian matrix as described above and estimate the ori-

entation tensor aij via Eq. (1), averaging over the completesub-volumes. Denote by k1 � k2 � k3 � 0 the eigenvaluesof the orientation tensor in descending order. Fisher et al.[43] use descriptors derived from these eigenvalues to classi-fy girdle and clustered distributions on the sphere: the shapec ¼ logðk1=k2Þ=logðk2=k3Þ and the strength n ¼ logðk1=k3Þ.A value of c below 1 indicates a girdle shape while c > 1is typical for clustered shapes. However, if the value of nis close to 0, then the directional distribution is isotropic or



Fig. 4. Glass fibre reinforced polymer. Sample and imaging: Institutfür Verbundwerkstoffe Kaiserslautern. Fibre width approximately10 lm, pixel size 3 lm, sample size approximately 1.8 · 3 · 3 mm3.Colours in the volume rendering indicate the preferred direction in thesub-volume, bright blue corresponds to the y-direction, dark blue tothe z-direction. (a) Volume rendering, (b) Slice through shading cor-rected image, (c) Diagonal elements of the orientation tensor, averagedover thick slices orthogonal to the x-axis.

(a)

(b)

(c)

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somehow neither clustered nor girdle-like. For the sub-vol-ume close to the welding seam we found c = 0.28, n = 1.65while c = 0.63, n = 1.98 away from the welding seam,hence both distributions are girdle-like. The latter is albeitmore concentrated, tending towards a clustered configura-tion.

4.4. Fibre felt: local thickness

A crucial input needed by the local directional analysis de-scribed above is the fibre thickness. For mechanical or flowproperties, local thickness of wall systems or fibre struc-tures is crucial. A volume weighted local thickness distribu-tion can be derived from the spherical granulometry. Thistransformation assigns to each pixel the size of the largestball inscribed in the structure and covering it. More pre-cisely, let N denote a random closed set in R

3, Br the ballof radius r centred at the origin in R

3, and N � Br the mor-phological opening of N with Br:

N � Br ¼[

fBr þ x : x 2 R3; Br þ x � Ng

The granulometry distribution function

GðdÞ ¼ 1� 1� VVðN � Bd=2Þ1� VVðNÞ

; d � 0

yields a volume weighted generalised size distribution. Itcan be estimated from the binary image N \ L\W by suc-cessive morphological openings with growing ballsN \ L\W � Bri until all foreground pixels have disap-peared. Finally, each pixel x is assigned the diameter ofthe ball used as structuring element in the step where x dis-appeared for the first time. The resulting image is called ul-timately opened [44] or granulometry image.

A more efficient algorithm is based on the medial axisMðN \ L\WÞ, which is the set of all centres of maximal in-scribed balls. That is, the medial axis coincides with thecrest lines of the EDT image. It should be noted that the ori-ginal structure can be reconstructed using the values of theEuclidean distance transform EDTðN \ L\WÞ in the pixelsforming the medial axis. One sorts these values in increas-ing order and respecting this order draws balls with radiusEDTðxÞ in all x 2 MðN \ L\WÞ, such that larger values

overwrite smaller ones. Pixel-wise multiplication by 2 thenyields the ultimate opening.

For fibrous structures, the granulometry distribution canbe interpreted as the volume weighted thickness distribu-tion. An example is the polymer felt used for dewateringpaper pulp, imaged using holotomography [45], and shownin Fig. 5. The histogram in Fig. 5 features two clear modescorresponding to the thin and thick fibres forming the felt’smicrostructure. A minus-sampling edge correction was ap-plied to remove too low thickness values at the image bor-ders. That is, only the thickness values 40 pixels away fromthe image borders have been taken into account.

A morphological opening with a ball, whose diameter islarger than the thickness of the thin fibres, yields the systemof the thick fibres, which can be obtained easily from thegranulometry image by global thresholding. On the otherhand, a white-top-hat transform (subtracting the opening re-sult from the original image) reveals the thin fibres. The lo-cal fibre directions can be estimated as described in Sec-tion 4.3 using the estimated thickness values for the thinand the thick fibres, respectively. Masking with the sepa-rated images yields the directional distributions for the thinand the thick fibres. The eigenvalue analysis of the orienta-tion tensor yields cthin = 1.43, nthin = 0.04 and cthick = 2.43,nthick = 0.88. The direction distribution of the thin fibres isthus neither clustered nor girdle-like while the direction dis-tribution of the thick fibres tends to be clustered.

4.5. Closed cell foam: correlation of constituents

For microstructures consisting of more than two constitu-ents, new tasks arise. One of those should answer the ques-tion whether two constituents are spatially correlated ornot: let N and W be macroscopically homogeneous randomclosed sets. If the two sets are independent, then

Pðx 2 N; y 2 WÞ ¼ Pðx 2 NÞPðy 2 WÞ for all x; y 2 R3

and deviation from this equality would indicate correlation.Frequently, N is the pore space of a porous microstructureand the other constituent is part of the solid structure. Thatis, instead of W we observe W 0 ¼ W \Nc, where Nc de-notes the closure of the complement of the pore system N.Consequently, the pore system N and W 0 will be correlatedin any case.



(a) (b) (c)

Fig. 5. Fibre felt, imaged by Helfen at the ESRF using the holotomography technique [45]. Pixel size 6.65 lm, sample size approximately(3.4 mm)3. The colours in slice (b) are assigned according to a rainbow colour table with blue corresponding to 0 pixels and red to the maximumthickness, which is approximately 35 pixels. The shaded part indicates the edge correction region. (a) Volume rendering, (b) Slice through colourcoded granulometry image, (c) Histogram of spherical granulometry distribution.

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If N and W are independent we have however

Pðx 2 N; y 2 W 0Þ ¼ Pðx 2 N; y 2 Nc; y 2 WÞ¼ Pðx 2 N; y 2 NcÞPðy 2 WÞ

for all x; y 2 R3: ð2Þ

Deviation from this necessary condition would thus indi-cate correlation. In order to verify it based on image data,let us first observe that

Pðx 2 N; y 2 WÞ � Pðx 2 NÞPðy 2 WÞ¼ Pðx 2 N; y 2 WÞ � VVðNÞVVðWÞ¼ covN;Wðx; yÞ

is the cross-covariance function of the sets N andW . Due tothe macroscopic homogeneity, it depends on the differenceh = y – x only:

covN;WðhÞ ¼Pðx 2 N; xþ h 2 WÞ � VVðNÞVVðWÞ; h 2 R

3 ð3ÞWe can now write the necessary condition (2) in terms ofthe cross-covariance function as covN;W 0ðhÞ ¼cov

N;NcðhÞVVðWÞ. We note that VVðWÞ ¼ VVðW 0Þ=

ð1� VVðNÞÞ and deduce the characteristic

tðhÞ ¼ covN;W 0ðhÞð1� VVðNÞÞcov

N;NcðhÞVVðW 0Þ

which can be estimated efficiently from the binary imagesof realisations of N and W 0 respectively, exploiting the dis-crete Fourier transform [46, 47]. The relationship t 1 isa necessary criterion for the sets N andW to be independent.

In the following, this criterion is used in order to examinethe pore initiation process in early metal foams when apply-ing the so-called powder foaming method where both thebulk material and the blowing agent in powder form aremixed, compacted, and subsequently heated. Here, we con-sider aluminium foams with titanium hydride (TiH2), whichwhen heated, releases hydrogen that in turn blows up themixture. In order to improve the blowing process and totightly control the resulting pore size distribution, a deeperunderstanding of the pore generation is needed.

To this end, aluminium foam samples quenched duringearly pore formation were imaged by synchrotron lCT[48, 49]. In the resulting images, the three components –metal matrix, blowing agent, and pores – are clearly visibledue to their well separated grey values, see Fig. 6. Thus,after a median filter for noise removal, global thresholdingyields the TiH2-component and the pores. An additionalpreprocessing step is necessary to avoid the physically im-possible \particles" within the pores, not connected to thematrix: labelling of the background, subsequent extractionof the largest component, and finally inverting the binaryimage is equivalent to filling the small holes, in the presentcase. As a side effect, there is a small number of pixels as-signed to TiH2 as well as to the pore space. These pixels



(a) (b) (c)

(d) (e)

Fig. 6. Slices through the (a) AlSi7 and (b) AW-6061 foam samples, volume renderings of the (c) AlSi7 and (d) AW-6061 foam samples, red–pore space, green–blowing agent leftovers. Images taken by L. Helfen at the ESRF, pixel size 0.7 lm, sample size approximately (4.6 mm)3.(e) Cross correlation.

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are removed from the pore space by a pixel-wise subtrac-tion.

Now, given binary images of the blowing agent and thepore system, we can tackle the actual analysis issue, i. e.whether the pores develop in the vicinity of the blowingagent particles or whether some other effect dominates theearly pore generation.

Let N and W denote the random closed sets of the poresand of TiH2, respectively. The blowing agent leftovers arecontained in the bulk only. Thus we observe W 0 ¼ W \Nc.It is exactly this kind of observable data for which the ne-cessary condition (2) in the case of independence of N andW holds. Figure 6 shows visualisations of foam samplesproduced using the two different precursor materials AW-6061 and AlSi7 together with estimates of t for the two ma-terials. For AW-6061, the strong deviation of t from 1 indi-cates short-range correlation of pores and blowing agent.For the AlSi7 samples, the pore initiation by the silica in-clusions dominates, thus there is no strong correlation be-tween pores and TiH2 (see [48] for details). Distance valuescorresponding to just one pixel, that is up to the length ofthe space diagonal of the lattice unit cell (in the present ex-ample 0.7 lm

ffiffiffi

3p

) are excluded from the analysis since pix-els at the interfaces between different constituents cannotbe assigned reliably due to the so-called halo or partial vol-ume effect: Grey values of these pixels are a weighted meanof the grey values of the neighbouring constituents.

If the restricting resource for the computation is memoryinstead of time, an alternative rationale can be used: If porespace and blowing agent were independent, then the obser-vable volume fraction of TiH2 should be constant, no matterat which distance from the pore system. This leads to thenecessary condition that the fraction of TiH2 observed inthe intersection of the bulk with stepwise dilations of thepore system N Br should be constant. Fast stepwise dila-tions by balls with growing radius can be achieved by sim-ple global thresholding of the Euclidean distance transformperformed on Nc. For details on this method see [47] or [25,Section 5.5.2].

5. Conclusions

Computed tomography combined with quantitative imageanalysis offers a wide variety of new possibilities to charac-terise microstructures. Neither the use cases chosen for thispaper nor the analysis methods presented cover exhaus-tively their respective ranges. The use cases do however re-present classes of structures – porous media and (fibre)composites – for which 3D imaging and image analysisare particularly useful and sought-after. The methods arerestricted to well-tested, general ones being applicable invarious applications. Nevertheless, the selection remainssubjective and incomplete. Missing for instance, are non-additive functionals, see e.g. [32] for a definition of a meandiameter of percolating pores, or further particle character-istics, cf. e. g. [50, 51].

Geometric characterisation discloses new avenues to re-construction of microstructures or to so-called virtual materi-al design. Starting from the characteristics measured in a 3Dimage, a stochastic geometric model is fitted to the micro-structure. Numerical simulation of properties such as thermalconductivity, elasticity, or filtration efficiency at the micro-scale and homogenisation can bridge the gap between micro-

scopic and macroscopic material properties. Selective alter-nation of the model parameters and therewith of the micro-structure allows study of the effect on the physicalproperties by simulating in realisations of the new micro-structure. This enables a deeper understanding of the relationbetween microstructure and material properties and even fos-ters optimisation of the microstructure with respect to the de-sired material property. Frequently used models are randomsystems of cylinders for fibres, random dense packings forgranular media and random Laguerre tessellations for foams,see [52] for an overview and [53–56] for examples.

The authors acknowledge support by (CM)2 Center for Mathemati-cal + Computational Modelling, by the German Research Foundation(DFG) under grant RE 3002/1-1 and by the German Federal Ministryof Education and Research through project 03MS603.

References

[1] D. Blavette, A. Bostel, J.M. Sarrau, B. Deconihout, A. Menand:Nature 363 (1993) 432.

[2] A. Kubis, G. Shiflet, R. Hull, D. Dunn: Metallurg. Mater. Trans.A 35 (2004) 1935.

[3] S.R. Stock: MicroComputed Tomography: Methodology and Ap-plications. CRC Press, Boca Raton (2008).

[4] J. Banhart (Ed.): Advanced Tomographic Methods in MaterialsResearch and Engineering. Oxford University Press (2008).

[5] J. Hsieh: Computed Tomography: Principles, Design, Artifactsand Recent Advances. SPIE Press, Bellingham, WA, 2nd edition(2009).

[6] A.C. Kak, M. Slaney: Principles of Computerized TomographicImaging. IEEE Press, New York (1988).

[7] M. Feser, J. Gelb, H. Chang, H. Cui, F. Duewer, S.H. Lau, A. Tka-chuk, W. Yun: Meas. Sci. Tech. 19 (2008) 094001.

[8] P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Lan-duyt, J.P. Guigay, M. Schlenker: Appl. Phys. Lett. 75 (1999) 2912.

[9] D. Chapman, W. Thomlinson, R.E. Johnston, D. Washburn, E. Pi-sano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, D. Sayers: Phys.Med. Biol. 42 (1997) 2015.

[10] T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni,P. Cloetens, E. Ziegler: Opt. Express 13 (2005) 6296.

[11] R.A. Brooks, G. Di Chiro: Med. Phys. 3 (1976) 237.[12] M. Di Michiel, J.M. Merino, D. Fernandez-Carreiras, T. Buslaps,

V. Honkimäki, P. Falus, T. Martins, O. Svensson: Rev. Sci. In-strum. 76 (2005) 043702.

[13] A. Rack, F. García-Moreno, C. Schmitt, O. Betz, A. Cecilia,A. Ershov, T. Rack, J. Banhart, S. Zabler: J. X-ray Sci. Tech. 18(2010) 429.

[14] A. King, G. Johnson, D. Engelberg, W. Ludwig, J. Marrow:Science 321 (2008) 382.

[15] G. Harding, J. Kosanetzky: Nucl. Instr. Meth. in Phys. Res. A 280(1989) 517.

[16] L. Helfen, A. Myagotin, A. Rack, P. Pernot, P. Mikulík, M. Di Mi-chiel, T. Baumbach: Phys. Status Solidi A 204 (2007) 2760.

[17] P. Bleuet, P. Cloetens, P. Gergaud, D. Mariolle, N. Chevalier,R. Tucoulou, J. Susini, A. Chabli: Rev. Sci. Instrum. 80 (2009)056101.

[18] A.M. Korsunsky, N. Baimpas, X. Song, J. Belnoue, F. Hofmann,B. Abbey, M. Xie, J. Andrieux, T. Buslaps, T. Khin Neo: ActaMater. 59 (2011) 2501.

[19] S.R. Stock, F. De Carlo, J.D. Almer: J. Struct. Biol. 161 (2008)144.

[20] N. Zoeger, C. Streli, P. Wobrauscheck, C. Jokubonis, G. Pepponi,P. Roschger, J. Hofstaetter, A. Berzelanovich, D. Wegrzynek,E. Chinea-Cano, A. Markowicz, R. Simon, G. Falkenberg: X-raySpectrom. 37 (2008) 3.

[21] M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C.M. Kewish,R. Wepf, O. Bunk, F. Pfeiffer: Nature 467 (2010) 436.

[22] A. Rack, S. Zabler, B.R. Müller, H. Riesemeier, G. Weidemann,A. Lange, J. Goebbels, M. Hentschel, W. Görner: Nucl. Instr. &Meth. in Phys. Res. A 586 (2008) 327.

[23] O. Wirjadi: Models and Algorithms for Image-Based Analysis ofMicrostructures. PhD thesis, Technical University Kaiserslautern(2009).



�20

12CarlH

anse

rVerlag,

Mun

ich,

German

ywww.ijmr.de

Not

forus

eininternet

orintran

etsites.Not

forelec

tron

icdistrib

ution.

[24] J. Ohser, W. Nagel, K. Schladitz: Image Anal. Stereol. 28 (2009)77.

[25] J. Ohser, K. Schladitz: 3d Images of Materials Structures – Pro-cessing and Analysis. Wiley VCH, Weinheim (2009).

[26] J. Ohser, W. Nagel, K. Schladitz, in: K.R. Mecke, D. Stoyan,(Eds.), Morphology of Condensed Matter, volume 600 of LectureNotes in Physics, Springer-Verlag, Berlin (2002) 275.

[27] N.R. Pal, S.K. Pal: Pattern Recogn, 26 (1993) 1277.[28] O. Wirjadi: Survey of 3d image segmentation methods, Technical

Report 123, Fraunhofer ITWM, Kaiserslautern (2007).[29] M. Sezgin, B. Sankur: J. Electron. Imaging 13 (2004) 146.[30] N. Otsu: IEEE T. Syst. Man Cyb. 9 (1979) 62.[31] P. Soille: Morphological image analysis, Springer-Verlag (1999).[32] J. Ohser, C. Ferrero, O. Wirjadi, A. Kuznetsova, J. Düll, A. Rack:

Int. J. Mater. Res. 103 (2012) 184, DOI:10.3139/146.110669[33] C.R. Maurer, V. Raghavan: IEEE T. Pattern Anal. 25 (2003) 265.[34] L. Vincent, P. Soille: IEEE T. Pattern Anal. 13 (1991) 583.[35] F.B. Tek, A.G. Dempster, I. Kale, in: C. Ronse, L. Najman, and

E. Decencire, (Eds.), Proc. Int. Symp. on Mathematical Morphol-ogy, volume 30 of Computational Imaging and Vision, Dordrecht,Springer-Verlag (2005) 441.

[36] M. Godehardt, K. Schladitz, in: Proceedings of the 9th EuropeanNDT Conference, Berlin (2006).

[37] D. Stoyan, W.S. Kendall, J. Mecke: Stochastic Geometry and ItsApplications, Wiley, Chichester, 2nd edition (1995).

[38] J. Ohser, C. Redenbach, K. Schladitz: Image Anal. Stereol. 28(2009) 101.

[39] J. Freitag, F. Wilhelms, S. Kipfstuhl: J. Glaciol. 50 (2004) 243.[40] R. Hilfer, in: K. R. Mecke, D. Stoyan (Eds.), Statistical Physics

and Spatial Statistics, volume 554 of Lecture Notes in Physics,Springer-Verlag, Heidelberg (2000) 203.

[41] H. Altendorf, D. Jeulin: Image Anal. Stereol. 28 (2009) 143.[42] M. Krause, J.M. Hausherr, B. Burgeth, C. Herrmann, W. Krenkel:

J. Mater. Sci. 45 (2010) 888.[43] N.I. Fisher, T. Lewis, B.J.J. Embleton: Statistical analysis of sphe-

rical data. Cambridge University Press, Cambridge (1987).[44] S. Beucher: Image Vision Comput. 25 (2007) 405.[45] P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache,

M. Schlenker: Proc. SPIE 4503 (2002) 82.[46] K. Koch, J. Ohser, K. Schladitz: Adv. Appl. Prob. 35 (2003) 603.[47] A. Rack, L. Helfen, T. Baumbach, S. Kirste, J. Banhart, K. Schla-

ditz, J. Ohser: J. Microscopy 232 (2008) 282.[48] L. Helfen, T. Baumbach, P. Cloetens, H. Stanzick, K. Schladitz,

J. Banhart: Appl. Phys. Lett. 86 (2005) 231907.[49] A. Rack, H.-M. Helwig, A. Bütow, A. Rueda, B. Matijasevic-Lux,

L. Helfen, J. Goebbels, J. Banhart: Acta Mater. 57 (2009) 4809.[50] E. Parra-Denis, C. Barat, D. Jeulin, C. Ducottet: Mater. Charact.

59 (2008) 338.

[51] I. Vecchio, K. Schladitz, G. Godehardt, M. Heneka, in: 3rd Inter-national Workshop: 3D Imaging, Analysis, Modeling and Simula-tion of Macroscopic Properties, Fontainebleau (2010).

[52] D. Jeulin, in: M. Bilodeau, F. Meyer, M. Schmitt (Eds.), Space,Structure and Randomness, volume 183 of Lecture Notes in Sta-tistics, Springer-Verlag, New York (2005) 183.

[53] H. Altendorf, D. Jeulin, in: 3rd International Workshop: 3D Imag-ing, Analysis, Modeling and Simulation of Macroscopic Proper-ties, Fontainebleau (2010).

[54] C. Peyrega, D. Jeulin, C. Delisée, J. Malvestio: Image Anal.Stereol. 28 (2009) 129.

[55] C. Redenbach, in: V. Capasso, G. Aletti, A. Micheletti (Eds.),Stereology and Image Analysis. Ecs10: Proc. 10th Europ. Conf.of ISS., volume 4 of The MIRIAM Project Series, ESCULAPIOPub. Co., Bologna (2009).

[56] K. Schladitz, S. Peters, D. Reinel-Bitzer, A.Wiegmann, J. Ohser:Comp. Mater. Sci. 38 (2006) 56.

(Received August 14, 2011; accepted November 14, 2011)

Bibliography

DOI 10.3139/146.110671Int. J. Mat. Res. (formerly Z. Metallkd.)103 (2012) 2; page 217–227# Carl Hanser Verlag GmbH & Co. KGISSN 1862-5282

Correspondence address

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Beyond imaging: on the quantitative analysis of tomographic volume data

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